Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x z w + x z t + y z t $ |
| $=$ | $x w t + x t^{2} + y t^{2}$ |
| $=$ | $x w^{2} + x w t + y w t$ |
| $=$ | $x^{2} w + x^{2} t + x y t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y - x^{2} y^{2} z - 60 x^{2} y z^{2} - 1125 x^{2} z^{3} + 225 y z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ -30x^{6} - 113x^{4} - 6750x^{2} + 12656 $ |
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This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:0:1)$, $(0:0:1:0:0)$, $(1:-1:1:0:0)$, $(0:1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{5}\cdot\frac{64000000000xt^{10}-949218750000y^{11}+1961718750000y^{9}w^{2}-10125000000000y^{9}wt+39905156250000y^{9}t^{2}-56948906250000y^{7}w^{2}t^{2}+18849003750000y^{7}wt^{3}-138087261000000y^{7}t^{4}+47353653000000y^{5}w^{2}t^{4}+177399860310000y^{5}wt^{5}-195015764862000y^{5}t^{6}+171877660422000y^{3}w^{2}t^{6}+169421819486640y^{3}wt^{7}+83735364541200y^{3}t^{8}-36911953125000yz^{8}t^{2}+154522856250000yz^{6}t^{4}+239040778860000yz^{4}t^{6}-25136671834320yz^{2}t^{8}-36209565794320yw^{2}t^{8}-25599690229024ywt^{9}-12342455023184yt^{10}+533935546875z^{11}+181933593750z^{9}w^{2}+4809375000000z^{9}wt+4867910156250z^{9}t^{2}+11094890625000z^{7}w^{2}t^{2}+11384516250000z^{7}wt^{3}+38660392125000z^{7}t^{4}+13133738250000z^{5}w^{2}t^{4}+2736117360000z^{5}wt^{5}+1478592198000z^{5}t^{6}+16693837164000z^{3}w^{2}t^{6}+16026284969040z^{3}wt^{7}+3244710686880z^{3}t^{8}-1017766258688zw^{2}t^{8}-7986719962624zwt^{9}-6203086886592zt^{10}}{t(2531250000y^{9}w-10631250000y^{9}t+6986250000y^{7}w^{2}t+14249250000y^{7}wt^{2}-9710550000y^{7}t^{3}+8627670000y^{5}w^{2}t^{3}+10834812000y^{5}wt^{4}+5837958000y^{5}t^{5}+309861360y^{3}w^{2}t^{5}+69289200y^{3}wt^{6}-2694782880y^{3}t^{7}+9175781250yz^{8}t+13655671875yz^{6}t^{3}-72159750yz^{4}t^{5}+2377521600yz^{2}t^{7}+1319822512yw^{2}t^{7}+920182992ywt^{8}+521386528yt^{9}-1423828125z^{9}w-1455468750z^{9}t+2071406250z^{7}w^{2}t+2107687500z^{7}wt^{2}-1558237500z^{7}t^{3}+789710625z^{5}w^{2}t^{3}+1504607625z^{5}wt^{4}+271180125z^{5}t^{5}-81111390z^{3}w^{2}t^{5}-36548580z^{3}wt^{6}+15671070z^{3}t^{7}+155474736zw^{2}t^{7}+422396064zwt^{8}+260693264zt^{9})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.72.3.d.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 5w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{15}t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{4}Y-X^{2}Y^{2}Z-60X^{2}YZ^{2}-1125X^{2}Z^{3}+225YZ^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.72.3.d.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -t$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 112x^{4}-75x^{2}wt-30x^{2}t^{2}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle x$ |
This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.
